Boolean operators

Boolean operators are used to connect and compare the relationship between . The result will be either TRUE or FALSE.

Boolean operation	Operator	Example
Both statements must be true for the argument as a whole to be true.	`AND`	`if x>=5 AND x <=20` Returns `TRUE` if `x` is any number between 5 and 20.
Only one of the statements needs be true for the argument as a whole to be true.	`OR`	`if x==2 OR x==5` Returns `TRUE` if `x` is either 2 or 5.
The opposite of the argument is true.	`NOT`	`if NOT(x==10)` Returns `TRUE` if `x` is not 10.
The argument is false if both statements are true. The argument is false if both statements are false. Otherwise the argument is true.	`XOR`	`if x<=10 XOR y<=10` Returns `TRUE` if one of `x` or `y` is greater than 10 and the other is not.

Boolean operation Both statements must be true for the argument as a whole to be true.

Operator

Boolean operation	Both statements must be true for the argument as a whole to be true.
Operator	`AND`
Example	`if x>=5 AND x <=20` Returns `TRUE` if `x` is any number between 5 and 20.

AND

Example

if x>=5 AND x <=20

Returns

TRUE

if

is any number between 5 and 20.

Boolean operation Only one of the statements needs be true for the argument as a whole to be true.

Operator

Boolean operation	Only one of the statements needs be true for the argument as a whole to be true.
Operator	`OR`
Example	`if x==2 OR x==5` Returns `TRUE` if `x` is either 2 or 5.

OR

Example

if x==2 OR x==5

Returns

TRUE

if

is either 2 or 5.

Boolean operation The opposite of the argument is true.

Operator

Boolean operation	The opposite of the argument is true.
Operator	`NOT`
Example	`if NOT(x==10)` Returns `TRUE` if `x` is not 10.

NOT

Example

if NOT(x==10)

Returns

TRUE

if

is not 10.

Boolean operation The argument is false if both statements are true. The argument is false if both statements are false. Otherwise the argument is true.

Operator

Boolean operation	The argument is false if both statements are true. The argument is false if both statements are false. Otherwise the argument is true.
Operator	`XOR`
Example	`if x<=10 XOR y<=10` Returns `TRUE` if one of `x` or `y` is greater than 10 and the other is not.

XOR

Example

if x<=10 XOR y<=10

Returns

TRUE

if one of

or

is greater than 10 and the other is not.

These can also be combined, for example:

If (x AND y) OR (NOT C)

Truth tables

are a way of showing all the possible outputs for inputs in a logical expression. , used in electronics and therefore computer circuits, are based on truth tables. More complex expressions can combine several inputs and outputs. In the following tables, 1 means TRUE and 0 means FALSE.

AND truth table

A	B	Output
0	0	0
0	1	0
1	0	0
1	1	1

A	0
B	0
Output	0

A	0
B	1
Output	0

A	1
B	0
Output	0

A	1
B	1
Output	1

In an AND truth table, the output is only TRUE if both inputs are also TRUE.

OR truth table

A	B	Output
0	0	0
0	1	1
1	0	1
1	1	1

A	0
B	0
Output	0

A	0
B	1
Output	1

A	1
B	0
Output	1

A	1
B	1
Output	1

In an OR truth table, the output is TRUE if any of the inputs are TRUE.

NOT truth table

A	Output
0	1
1	0

A	0
Output	1

A	1
Output	0

A NOT truth table reverses the input value.

XOR truth table

A	B	Output
0	0	0
0	1	1
1	0	1
1	1	0

A	0
B	0
Output	0

A	0
B	1
Output	1

A	1
B	0
Output	1

A	1
B	1
Output	0

In an XOR truth table, the output is TRUE if only one of the input values is TRUE.

Boolean notation

Logical expressions can also be expressed using Boolean algebra notation. Instead of writing the words AND, OR, NOT or XOR it is often written using the following shorthand notation.

Boolean expression	Shorthand notation
A AND B	A . B
A OR B	A + B
NOT A	A’ (can be also shown as Ā)
A XOR B	A ⊕ B

Boolean expression	A AND B
Shorthand notation	A . B

Boolean expression	A OR B
Shorthand notation	A + B

Boolean expression	NOT A
Shorthand notation	A’ (can be also shown as Ā)

Boolean expression	A XOR B
Shorthand notation	A ⊕ B

Boolean identities and rules

Rule	AND form	OR form
Commutative law	A . B = B . A	A + B = B + A
Associative law	(A . B) . C = A . (B . C)	(A + B) + C = A + (B + C)
Distributive law	(A . B) + C = (A + C) . (B + C)	(A + B) . C = (A . C) + (B . C)
Identity law	A . 1 = A	A + 0 = A
Zero and one law	A . 0 = 0	A + 1 = 1
Inverse law	A . A’ = 0	A + A’ = 1
Idempotent law	A . A = A	A + A = A
Absorption law	A . (A + B) = A	A + A . B = A A + A’ B = A + B
Double complement law	Ā = A

Rule	Commutative law
AND form	A . B = B . A
OR form	A + B = B + A

Rule	Associative law
AND form	(A . B) . C = A . (B . C)
OR form	(A + B) + C = A + (B + C)

Rule	Distributive law
AND form	(A . B) + C = (A + C) . (B + C)
OR form	(A + B) . C = (A . C) + (B . C)

Rule	Identity law
AND form	A . 1 = A
OR form	A + 0 = A

Rule	Zero and one law
AND form	A . 0 = 0
OR form	A + 1 = 1

Rule	Inverse law
AND form	A . A’ = 0
OR form	A + A’ = 1

Rule	Idempotent law
AND form	A . A = A
OR form	A + A = A

Rule	Absorption law
AND form	A . (A + B) = A
OR form	A + A . B = A A + A’ B = A + B

Rule	Double complement law
AND form	Ā = A

Examples:

A AND A = A

How?

If A is 0, then the expression is 0 AND 0 = 0, which is equal to A.

If A is 1, then the expression is 1 AND 1 = 1, which is equal to A.

A OR A = A

How?

If A is 0, then the expression is 0 OR 0 = 0, which is equal to A.

If A is 1, then the expression is 1 OR 1 = 1, which is equal to A.

A AND NOT A = 0

How?

If A is 1, then NOT A = 0.

Then the expression becomes 1 AND 0 = 0.

If A is 0, then NOT A = 1.

Then the expression becomes 0 AND 1 = 0.

Simplifying Boolean expressions

You may be asked to simplify a Boolean expression, for example:

X = A AND B OR A AND NOT B

X = A AND (B OR NOT B)

X = A AND 1

X = A

Data representation - EduqasBoolean operators